| import OAI.Probability.MatroidProphet.Pivots.Completeness |
| import Mathlib.Data.Finset.Sort |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.Pivots |
|
|
| open Set Finset |
|
|
| variable {α β γ O A : Type*} [Fintype α] |
|
|
| lemma isPivot_reindex |
| {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Fintype α] |
| (M : Matroid α) (label : β → α) (time : β → ℕ) |
| (e : γ ≃ β) (f : α) (o : γ) : |
| IsPivot M (label ∘ e) (time ∘ e) f o ↔ IsPivot M label time f (e o) := by |
| have himage (p : β → Prop) : (label ∘ e) '' {x | p (e x)} = label '' {x | p x} := by |
| ext a |
| constructor |
| · rintro ⟨x, hx, rfl⟩ |
| exact ⟨e x, hx, rfl⟩ |
| · rintro ⟨x, hx, rfl⟩ |
| exact ⟨e.symm x, by simpa using hx, by simp⟩ |
| unfold IsPivot |
| simp only [Function.comp_apply] at himage ⊢ |
| rw [himage (fun x => time x ≤ time (e o)), himage (fun x => time x < time (e o))] |
|
|
| noncomputable def freeMarkedPatterns [Fintype O] [LinearOrder O] [Fintype A] |
| (M : Matroid α) (oldLabel : O → α) (movableLabel : A → α) |
| {n : ℕ} (test : Fin n → α) (oldMark : O → Bool) : Finset (Finset (Fin n)) := by |
| classical |
| exact Finset.univ.filter fun t => ∃ time : O ⊕ A → ℕ, |
| StrictMono (fun o => time (Sum.inl o)) ∧ Function.Injective time ∧ |
| ∃ mark : A → Bool, ∀ f, ∃ o, |
| IsPivot M (Sum.elim oldLabel movableLabel) time (test f) o ∧ |
| (Sum.elim oldMark mark o = true ↔ f ∈ t) |
|
|
| lemma card_freeMarkedPatterns_le [Fintype O] [LinearOrder O] [Fintype A] |
| (M : Matroid α) (d : α) (oldLabel : O → α) (movableLabel : A → α) |
| {n : ℕ} (test : Fin n → α) (oldMark : O → Bool) |
| (s : ℕ) (hs : Fintype.card A ≤ s) |
| (hground : ∀ o, Sum.elim oldLabel movableLabel o ∈ M.E) : |
| (freeMarkedPatterns M oldLabel movableLabel test oldMark).card ≤ |
| 4 ^ s * ∑ k ∈ Finset.Iic s, n.choose k := by |
| classical |
| let eO := Fintype.orderIsoFinOfCardEq O rfl |
| let eA := (Fintype.equivFin A).symm |
| let e := Equiv.sumCongr eO.toEquiv eA |
| let b : ℕ → α := fun k => if hk : k < Fintype.card O then oldLabel (eO ⟨k, hk⟩) else d |
| let a : Fin (Fintype.card A) → α := movableLabel ∘ eA |
| let m : Fin (Fintype.card O) → Bool := oldMark ∘ eO |
| have hlabel : occurrenceLabel b a = Sum.elim oldLabel movableLabel ∘ e := by |
| funext o |
| cases o <;> simp [occurrenceLabel, b, a, e] |
| have hsub : freeMarkedPatterns M oldLabel movableLabel test oldMark ⊆ |
| markedPivotPatterns M b a test m := by |
| intro t ht |
| obtain ⟨time, horder, htime, mark, hrecord⟩ := (Finset.mem_filter.mp ht).2 |
| refine Finset.mem_filter.mpr ⟨Finset.mem_univ _, time ∘ e, ?_, htime.comp e.injective, |
| mark ∘ eA, ?_⟩ |
| · intro i j hij |
| exact horder (eO.strictMono hij) |
| · intro f |
| obtain ⟨o, hp, hm⟩ := hrecord f |
| refine ⟨e.symm o, ?_, ?_⟩ |
| · rw [hlabel] |
| exact (isPivot_reindex M (Sum.elim oldLabel movableLabel) time e (test f) (e.symm o)).2 |
| (by simpa using hp) |
| · cases o <;> simpa [occurrenceMark, e, m, Function.comp_def] using hm |
| apply (Finset.card_le_card hsub).trans |
| apply card_markedPivotPatterns_le M b a test m s hs |
| intro o |
| rw [hlabel] |
| exact hground (e o) |
|
|
| end MatroidProphet.Pivots |
|
|
| end OAI |
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|
